365 research outputs found
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Constructing dense graphs with sublinear Hadwiger number
Mader asked to explicitly construct dense graphs for which the size of the
largest clique minor is sublinear in the number of vertices. Such graphs exist
as a random graph almost surely has this property. This question and variants
were popularized by Thomason over several articles. We answer these questions
by showing how to explicitly construct such graphs using blow-ups of small
graphs with this property. This leads to the study of a fractional variant of
the clique minor number, which may be of independent interest.Comment: 10 page
On Routing Disjoint Paths in Bounded Treewidth Graphs
We study the problem of routing on disjoint paths in bounded treewidth graphs
with both edge and node capacities. The input consists of a capacitated graph
and a collection of source-destination pairs . The goal is to maximize the number of pairs that
can be routed subject to the capacities in the graph. A routing of a subset
of the pairs is a collection of paths such that,
for each pair , there is a path in
connecting to . In the Maximum Edge Disjoint Paths (MaxEDP) problem,
the graph has capacities on the edges and a routing
is feasible if each edge is in at most of
the paths of . The Maximum Node Disjoint Paths (MaxNDP) problem is
the node-capacitated counterpart of MaxEDP.
In this paper we obtain an approximation for MaxEDP on graphs of
treewidth at most and a matching approximation for MaxNDP on graphs of
pathwidth at most . Our results build on and significantly improve the work
by Chekuri et al. [ICALP 2013] who obtained an approximation
for MaxEDP
Approximation Algorithms for Connected Maximum Cut and Related Problems
An instance of the Connected Maximum Cut problem consists of an undirected
graph G = (V, E) and the goal is to find a subset of vertices S V
that maximizes the number of edges in the cut \delta(S) such that the induced
graph G[S] is connected. We present the first non-trivial \Omega(1/log n)
approximation algorithm for the connected maximum cut problem in general graphs
using novel techniques. We then extend our algorithm to an edge weighted case
and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark
contrast to the classical max-cut problem, we show that the connected maximum
cut problem remains NP-hard even on unweighted, planar graphs. On the positive
side, we obtain a polynomial time approximation scheme for the connected
maximum cut problem on planar graphs and more generally on graphs with bounded
genus.Comment: 17 pages, Conference version to appear in ESA 201
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